On the Complexity of Role Colouring Planar Graphs, Trees and Cographs

TL;DR

This paper investigates the computational complexity of role colouring across different graph classes, proving NP-hardness for planar graphs, polynomial solvability for trees under certain conditions, and polynomial algorithms for cographs.

Contribution

It establishes complexity results for role colouring in planar graphs, trees, and cographs, including NP-hardness, polynomial solvability, and constructive algorithms.

Findings

NP-hardness of role colouring in planar graphs

Polynomial algorithms for trees with specific parameters

Cographs are always k-role-colourable with polynomial construction

Abstract

We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with $1< k <n$ colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as $k$ is either constant or has a constant difference with $n$, the number of vertices in the tree. Finally, we prove that cographs are always $k$-role-colourable for $1<k\leq n$ and construct such a colouring in polynomial time.